We can use SWAP gates, identity gates, and some rearrangement to embed the
circuits from Theorem REF  into \(D\)  spacetime dimensions so that all
gates become spatially local.
The codes constructed in this way are not just sparse, but also geometrically
local. This results in nearly optimal distances of
\(\Omega \bigl (n^{1-1/D-\epsilon }\bigr )\)  [1]}, as well as spatially
local codes that achieve \(kd^{2/(D-1)} \ge \Omega (n)\)  in \(D=4\)  dimensions.
